Dynamics of O(2) excitations in a non-reciprocal medium

Abstract

We investigate emergent dynamics due to non-reciprocity in the $\mathcal{O}(2)$ model. The lattice XY model, where non-reciprocity stems from vision cone like couplings, can be described by a continuum description in which non-reciprocity translates into a new term depending on the rotational of the orientation field. We argue that non-reciprocity is akin to activity and we highlight the connection between our hydrodynamic equation and the constant density Toner-Tu framework. The active force advects and reshapes patterns, a generic feature found in many non-reciprocal systems. We show how $1d$ excitations in the non-reciprocal $\mathcal{O}(2)$ model can be described by a generalized Burgers equation, derived from our continuum model. We then extend the results to $2d$ perturbations. As such, we establish the first principles of excitation trajectory control in a non-reciprocal $\mathcal{O}(2)$ medium. Concretely, we explain how tuning the degree of non-reciprocity and the orientation of the background medium impacts the time evolution of excitations. We also showcase how initially different excitations lead to very different dynamical behavior. Non-reciprocity also affects the stability of defect-free excitations with non-zero winding numbers and, unlike in its equilibrium $O(2)$ counterpart, enables the system, above a certain threshold, to relax to its ground state.

Figures (11)

• Figure 1: Three possible kernels describing the angular dependence of the coupling strength: (a) a uniform kernel (b) a sharp vision cone, taking value 1 in front and 0 behind. Here, the vision cone aperture is $\Theta=2\pi-2$ . (c) a smooth vision cone, smoothly changing from $1+\sigma$ (in front) to $1-\sigma$ (behind), here with $\sigma = 0.6$. All the kernels we study are centered on the current orientation of the spin, here depicted by the horizontal arrows. The dotted lines in panels (b, c) represent the uniform kernel in (a). (d) The three kernels illustrated in (a)-(c) as functions of $\varphi_{ij}=\text{Angle}(\boldsymbol{\hat{S}}_i\cdot \boldsymbol{u}_{ij})$. (e) Illustration of the impact of a positive rotational vector field on an individual spin; see main text for discussion.
• Figure 2: Time evolution of an initial $1d$ gaussian excitation. We show a partial rectangular window of the system. The orientation field $\theta$ is plotted in panels (a-c) for $t/\tau = 0, 0.4, 1$ respectively. The other parameters are $\alpha =100,\, \bar{\sigma}/\sqrt{K\alpha} = 20,\, \theta_0=0$. The initial profile travels to the left (black arrow) and develops a front/back asymmetry (smoother at the front, sharper at the back).
• Figure 3: (a) Propagation of a perturbation $\theta=\delta(x)$ across the system over time. Parameters: small non-reciprocity $\bar{\sigma}/\sqrt{K\alpha} = 2, \alpha = 100, t_{\text{max}}/\tau = 20, \theta_0=0$. Inset: rescaled data $\theta\sqrt{t/\tau}$ as a function of $(x-x_{\text{peak}})/(L\sqrt{t/\tau})$. (b) Same but for larger non-reciprocity $\bar{\sigma}/\sqrt{K\alpha} = 20$, and shorter simulation time $t_{\text{max}}/\tau = 2.5$. Inset: rescaled data $\theta \,(t/\tau)^{1/3}$ as a function of $(x-x_{\text{peak}})/(L(t/\tau)^{1/3})$. (c) A gaussian profile of height $\delta_0 = \pi/2$ and background orientation $\theta_0 = 0$. Colors indicate the orientation $\theta$, see color map. The horizontal gray arrows represent $-\cos\theta$, which physically corresponds to the local convective velocity, see main text for details. The symbols at the top of the boxes are used in the main text to guide the reader. (d) Peak velocity for $\theta_0=0$ (blue, $v<0$) and $\theta_0=\pi/2$ (pink, $v>0$). The dashed and dotted lines correspond to Eq. (\ref{['eq:sigmacos']}). Dashed line: $\beta = 1/3, c= 0.72$. Dotted line: $\beta = 1/2, c= 0.62$. Inset: steady peak velocity $v_\infty$ against $\theta_0$, for $\delta_0 = \pi/20$. The back dash line perfectly matching the data is $-\cos \theta_0$.
• Figure 4: (a) Profile $\theta(x) = \pi /2 + \delta$, where $\delta$ is given by Eq. (\ref{['eq:sinx_1d']}) with $\delta_0>0$. Different colors are different times on a logarithmic scale, increasing from yellow to blue (steady state). The same color scale is used in panels (a,b,d,e). The vertical grey lines are plotted to guide the eye and have the same value in all panels (a,b,d,e). The arrows represent the direction of motion of the two parts of the perturbation. (b) Same plot for the squared magnitude $S^2(x)$. It dips at $x=0$ where the derivative $\partial\theta/\partial x$ is the largest. (c) The relative amplitude $\delta_\text{peak}/|\delta_0|$ of a perturbation with $\delta_0>0$ as a function of time for different values of non-reciprocity. The initial value $\delta_\text{peak}/\delta_0 = 0.523$. The dashed and solid lines indicate $\sim t^{-1/2}$ and $\sim t^{-1}$, respectively. (d-f) Same quantities but for $\delta_0<0$, swapping the locations of the two peaks. In panel (f), the solid and dotted lines indicate $\sim t^{-1}$ and $\sim t^{-3}$, respectively.
• Figure 5: Time evolution of an initial isotropic $2d$ gaussian perturbation, with $\theta_0 = 0, \delta_0 =\pi/2$. We show a partial rectangular window of the system. The orientation field $\theta$ is plotted in panels (a-c) for $t/\tau = 0, 0.6, 1.2$ respectively. The dots track the past positions of the peak. Other parameters: $\alpha =100, \bar{\sigma}/\sqrt{K\alpha} = 20$. (d-f) cross-sections of the orientational field $\theta$ along a vertical (dash, against $y/L$) and horizontal (solid, against $x/L$) lines passing through the peak of the perturbation in panels (a-c).